Math4201 Topology I (Lecture 20)

Quotient topology

More propositions

Proposition for quotient maps in restrictions

Let $X,Y$ be topological spaces and $p:X\to Y$ is surjective and open/closed. Let $A\subseteq X$ be saturated by $p$ , ( $p^{-1}(p(A))=A$ ).

Then $q: A\to p(A)$ given by the restriction of $p$ is open/closed surjective map (In particular, it’s a quotient map).

Proof

$q$ is surjective and continuous. Now assume $p$ is open and we will show that $q$ is also open. Any open subspace of $A$ is given as $U\cap A$ where $U$ is open in $X$ . By definition, $q(U\cap A)=p(U\cap A)=p(U)\cap p(A)$

To see the second identity:

$p(U\cap A)\subseteq p(U)\cap p(A)$

$\forall y\in p(U\cap A)$ , $y=p(x)$ with $x\in U\cap A$ , since $x\in A$ and $x\in U$ , $y=p(x)\in p(U)\cap p(A)$

$p(U)\cap p(A)\subseteq p(U\cap A)$

$\forall y\in p(U)\cap p(A)$ , $y=p(x_1)$ with $x_1\in U$ and $y=p(x_2)$ with $x_2\in A$ , since $x_1\in U$ and $x_2\in A$ , $y=p(x_1)=p(x_2)\in p(U\cap A)$

So $x_1=x_2\in U\cap A$ , $y=p(x_1)=p(x_2)\in p(U)\cap p(A)$ , $y\in p(U\cap A)$ .

Note that $p(U)\subseteq X$ is open by $p$ is an open map.

So $p(U)\cap p(A)$ is open in $p(A)$ .

$q(U\cap A)=p(U\cap A)=p(U)\cap p(A)$ is open.

So $q$ is open in $p(A)$ .

Simplicial complexes (extra chapter)

Definition for simplicial complexes

Simplicial complexes are topological space with simplices ( $n$ dimensional triangles) as their building blocks.

Definition for n dimensional simplex

Let $v_0,\dots,v_n$ be points in $\mathbb{R}^m$ such that $v_n-v_0$ , $v_{n-1}-v_0$ , $\cdots$ , and $v_1-v_0$ are linearly independent in $\mathbb{R}^m$ . (in particular $n\leq m$ ).

The $n$ -dimensional simplex determined by $\{v_0,\dots,v_n\}$ is given as:

\Delta^n\coloneqq [v_0,\dots,v_n]=\{t_0v_0+t_1v_1+\cdots+t_nv_n\vert t_i\geq 0, \sum_{i=0}^n t_i=1\}

The coefficients $t_0,\dots,t_n$ are called barycentric coordinates.

Example of simplicial complex

$n=0$ ,

$\Delta^0=\{v_0\}$

$n=1$ ,

$\Delta^1=\{t_0v_0+t_1v_1\vert t_0+t_1=1\}$ , this is the line segment between $v_0$ and $v_1$ .

$n=2$ ,

$\Delta^2=\{t_0v_0+t_1v_1+t_2v_2\vert t_0+t_1+t_2=1\}$ , this is the triangle with vertices $v_0,v_1,v_2$ .

Note

Every non-empty subset $\{v_{i_0},\dots,v_{i_k}\}$ of $\{v_0,\dots,v_n\}$ determines a $k$ dimensional simplex $[v_{i_0},\dots,v_{i_k}]\subseteq \Delta^n=[v_0,\dots,v_n]$ . Inside the $n$ dimensional simplex $t_{i_0}v_{i_0}+\cdots+t_{i_n}v_{i_k}\in \Delta^n$ . Where the coefficient $t_j$ of $v_j\notin \{v_{i_0},\dots,v_{i_n}\}$ is $0$ .

Any such $k$ dimensional simplex is called a face of the simplex $[v_{i_0},\dots,v_{i_n}]$ .

Example of faces for simplicial complex

For a triangle $[v_0,v_1,v_2]$ , the faces are $[v_0,v_1]$ , $[v_0,v_2]$ , and $[v_1,v_2]$ (the edges of the triangle).

Definition for abstract simplicial complex

Let $V$ be a finite or countable set, an abstract simplicial complex on $V$ is a collection of finite non-empty subset of $V$ , denoted by $K$ . And the two conditions are satisfied:

If $\sigma\in K$ and $\tau\subseteq \sigma$ , then $\tau\in K$ .
For any $v\in V$ , $\{v\}\in K$ .

Example of abstract simplicial complex

Let $V=\{a,b,c,d\}$ .

If we want to include $\{a,b,c\}$ , then we need to include $\{a,b\}$ and $\{b,c\}$ , so we have $K=\{\{a,b,c\},\{a,b\},\{b,c\},\{a\},\{b\},\{c\},\{d\}\}$ is an abstract simplicial complex.

Topological realization of abstract simplicial complex

Let $\bigsqcup_{\sigma\in K}\Delta^{|\sigma|-1}$ be the disjoint union of all $|\sigma|-1$ dimensional simplices in $K$ .

\tilde{X_k}=\bigsqcup_{\sigma\in K}\Delta^{|\sigma|-1}

We use subspace topology to define a topology on $\Delta^n$ and the union of such topology for each $\Delta^{|\sigma|-1}$ defines a topology on $\tilde{X_k}$ .

We define the equivalence relation $x\in \Delta_{\sigma}^{|\sigma|-1}\sim x'\in \Delta_{\sigma'}^{|\sigma'|-1}$ if $x\in \Delta_{\sigma'\cap \sigma}^{|\sigma'\cap \sigma|-1}\subseteq \Delta_{\sigma}^{|\sigma|-1}$ . and $x'\in \Delta_{\sigma'\cap \sigma}^{|\sigma'\cap \sigma|-1}\subseteq \Delta_{\sigma'}^{|\sigma'|-1}$ .

are the sample points of $\Delta_{\sigma\cap \sigma'}^{|\sigma\cap \sigma'|-1}$ .

$X_K$ is the quotient space of $\tilde{X_k}$ by the equivalence relation.

Continue next time.